Distinguishing pure representations by normalized traces

TL;DR

This paper proves that pure Galois representations with equal normalized traces are related by a simple twist, extending recent results and deepening understanding of their structure.

Contribution

It establishes a criterion for when two pure Galois representations are related by a twist based on normalized traces, generalizing previous work.

Findings

Normalized traces determine Frobenius-semisimplifications up to twist.

Pure representations with equal normalized traces are twists of each other.

The result extends the analogy with Patankar and Rajan's recent findings.

Abstract

Given two pure representations of the absolute Galois group of an $\ell$-adic number field with coefficients in $\overline{\mathbb{Q}}_p$ (with $\ell\neq p$), we show that the Frobenius-semisimplifications of the associated Weil--Deligne representations are twists of each other by an integral power of certain unramified character if they have equal normalized traces. This is an analogue of a recent result of Patankar and Rajan in the context of local Galois representations.